How can we prove that a Pythagorean triple is primitive?

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Discussion Overview

The discussion revolves around how to demonstrate that a given Pythagorean triple, specifically of the form (a²-b², 2ab, a²+b²), is primitive, meaning that the greatest common divisor (gcd) of the three numbers is 1. Participants explore various conditions and methods for proving this property.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants assert that for the triple to be primitive, the integers a and b must be coprime (gcd(a,b)=1) and of opposite parity (one even, one odd).
  • One participant suggests that a contradiction argument could be useful, proposing to assume that the numbers x, y, z have a common factor d greater than 1 and questioning if d can be assumed to be prime.
  • Another participant reiterates the need to show that gcd(x,y,z)=1 but does not provide a specific method for doing so.

Areas of Agreement / Disagreement

Participants generally agree on the conditions involving a and b for the triple to be primitive, but there is no consensus on the specific method to prove that gcd(x,y,z)=1.

Contextual Notes

Some assumptions regarding the properties of a and b, such as their parity and coprimality, are mentioned but not fully explored or proven within the discussion.

Poirot1
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I know that (a^2-b^2,2ab,a^2+b^2) is pythagorean triple. How to show it is primitive? i.e
gcd(x,y,z)=1
 
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Re: primitive pythagorean triple

Hello, Poirot!

I know that (x,y,z) = (a2 - b2, 2ab, a2 + b2) is a Pythagorean triple.

How to show it is primitive? .i.e. gcd(x,y,z) = 1
I'm not sure how we "show" it, but here is a fact.

For a primitive Pythagorean triple, a and b must be of opposite parity.
. . (One must be even, the other must be odd.)
 
Re: primitive pythagorean triple

Poirot said:
I know that (a^2-b^2,2ab,a^2+b^2) is pythagorean triple. How to show it is primitive? i.e
gcd(x,y,z)=1
The conditions for the triple to be primitive are that gcd(a,b)=1 and a, b are of opposite parity. See Pythagorean triple - Wikipedia, the free encyclopedia.
 
Re: primitive pythagorean triple

I'm pretty sure a contradiction argument is expedient.

assume x,y,z have a common factor d not equal to 1.
Since everything can be factorized into primes can I assume d is prime?
 

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